The fiber of this morphism \(M_v \mathbf {a}_v^{-1}(\mathcal {O}_J, 0)\)  will be our main object of interest in this paper. The points of \(M_v\)  parameterize \(S\) -equivalence classes of semi-stable sheaves, or equivalently isomorphism classes of polystable sheaves (see [1]}), on \(J\)  having determinant bundle isomorphic to \(\mathcal {O}_J(2\theta )\)  and second Chern class summing up to 0.
